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Laser Physics

Bose-Einstein condensate in a double well potential in the vicinity of a critical point

Barbara Oleś, Paweł Zin, Jan Chwedeńczuk, Krzysztof Sacha, Marek Trippenbach

Published Feb 2, 2010• 12 citations
Publisher

Abstract

We consider a Bose-Einstein condensate with an attractive particle interaction in a symmetric double well potential. It is known that in the mean field description a symmetric solution for the condensate wavefunction becomes unstable when the interactions are stronger than some critical value. We analyze the system around the critical point applying the number conserving Bogoliubov theory and exact diagonalization of the two mode hamiltonian. It allows for estimation of the density fluctuations in the system. Fluctuations of the order parameter, which is defined in the mean field description, turn out to be maximal at the critical point.

DOI
10.1134/s1054660x10050130

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We consider a Bose-Einstein condensate with an attractive particle interaction in a symmetric double well potential. It is known that in the mean field description a symmetric solution for the condensate wavefunction becomes unstable when the interactions are stronger than some critical value. We analyze the system around the critical point applying the number conserving Bogoliubov theory and exact diagonalization of the two mode hamiltonian. It allows for estimation of the density fluctuations in the system. Fluctuations of the order parameter, which is defined in the mean field description, turn out to be maximal at the critical point.
We consider a Bose gas with an attractive interaction in a symmetric double-well potential. In the mean-field approximation, the ground-state solution spontaneously breaks the symmetry of the trapping potential above a certain value of the interaction strength. We demonstrate how the Landau-Ginzburg scheme of the second-order phase transition emerges from the quantum model and show the link to the spontaneous symmetry breaking mentioned above. We identify the order parameter, the critical point and analyze quantum fluctuations around it.
We employ mean-field, Bogoliubov and many-body theories to study critical fluctuations in the position and momentum of a Bose-Einstein condensate whose translation symmetry is spontaneously broken due to attractive interactions. In a homogeneous system, the many-body ground state of the symmetry-preserving Hamiltonian is very fragile against superposition of low-lying states, while the mean-field theory predicts a stable bright soliton which spontaneously breaks translation symmetry. We show that weak symmetry-breaking perturbations cause the translation-symmetric many-body ground state to cross over to a many-body bright soliton. We argue that the center-of-mass fluctuations in the soliton state arise primarily from the depletion of the condensate to translation modes. We develop an extended mean-field theory to analytically reproduce these results obtained by the exact diagonalization method.
We consider a homogeneous Bose gas of particles with an attractive interaction in an elongated three-dimensional box with periodic boundary conditions. Mean-field theory predicts for this system a spontaneous breaking of the translational symmetry at a certain value of the interaction strength. We show that at this point a second-order quantum phase transition occurs. We investigate the system in the vicinity of the critical point using Bogoliubov theory and a continuous description, that allows us to analyze quantum fluctuations in the system even when the Bogoliubov approach breaks down.
A Yukawa-like equation, introduced to evaluate the first order correlation function in a Bose-Einstein condensate, can be used to evaluate the atomic density fluctuations. It does not require a few mode approximation. We apply this approach to calculate the atomic fluctuations of a condensate of attractive atoms in a double-well potential. The atomic system undergoes a spatial symmetry breaking above a critical number Nc{N}_{c}Nc​ of atoms. Near Nc{N}_{c}Nc​ the atomic fluctuation $\ensuremath{\delta}N$ inside a single well becomes very large and experimentally observable by a phase-contrast imaging technique. $\ensuremath{\delta}N$ is related to the collective mode temperature, which may be different from the condensate temperature measured by means of the thermal tails. A one-dimensional array of wells is also considered.
We consider the quantum dynamics of a neutral atom Bose- Einstein condensate in a double-well potential, including hard-sphere particle interactions. Using a mean-field factorization we show that the coherent oscillations due to tunnelling are suppressed when the number of atoms exceeds a critical value. An exact quantum solution, in a two-mode approximation, shows that the mean-field solution is modulated by a quantum collapse and revival sequence. Chaotic dynamics results when the potential is modulated.